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58: Bringing Curvy Back (Gaussian Curvature)
539 Plays4 years ago
In introductory geometry classes, many of the objects dealt with can be considered 'elementary' in nature; things like tetrahedrons, spheres, cylinders, planes, triangles, lines, and other such concepts are common in these classes. However, we often have the need to describe more complex objects. These objects can often be quite organic, or even abstract in shape, and include things like spirals, flowery shapes, and other curved surfaces. These are often described better by differential geometry as opposed to the more elementary classical geometry. One helpful metric in describing these objects is how they are curved around a certain point. So how is curvature defined mathematically? What is the difference between negative and positive curvature? And what can Gauss' Theorema Egregium teach us about eating pizza?
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[Featuring: Sofía Baca, Meryl Flaherty]
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Transcript
Introduction to Geometry and Shapes
00:00:00
Speaker
In introductory geometry classes, many of the objects dealt with can be considered elementary in nature. Things like tetrahedrons, spheres, cylinders, planes, lines, and other such concepts are common in these classes and are part of classical geometry.
00:00:16
Speaker
However, we often have the need to describe more complex objects. These objects can be quite organic or even abstract in shape, and include things like spirals, flowery shapes, and other curved surfaces. These are often described better by differential geometry as opposed to the more elementary classical geometry. One helpful metric in describing these objects is how they are curved around a certain point. So how is curvature defined mathematically?
00:00:41
Speaker
What is the difference between a negative and a positive curvature?
Gauss's Theorem and Everyday Life
00:00:44
Speaker
And what can Gauss's Theoremma Egrigium teach us about eating pizza? All this and more on this episode of Breaking Math.
Meet the Hosts and Episode Overview
00:00:51
Speaker
Episode 58, Bringing Curvy Back.
00:00:59
Speaker
I'm Sophia. And I'm Meryl. And you're listening to Breaking Math. Meryl's been on quite a bit recently. She was on what episodes were you on recently? I was on the one about spheres and the one about bases.
00:01:13
Speaker
Yeah, and Meryl is actually going to be part more of Breaking Math now, and she's going to be a co-host as well. So this is sort of a welcome of Meryl to that part of the show. Before we continue, let's do some plugs.
Supporting the Show
00:01:27
Speaker
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00:01:32
Speaker
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00:01:55
Speaker
And it's under construction right now, but it's still usable. And if you want to email us, ideas, comments, questions, or corrections, you can email us at breakingmathpodcast at gmail.com. And if you want to listen to something quite like Breaking Math, you could list to our sister podcast, Turning Rabbit Holes.
Classical vs Differential Geometry
00:02:13
Speaker
So Meryl, do you want to walk through the difference between classical and differential geometry?
00:02:17
Speaker
So what I would say is we're talking about with more classical geometry, think Euclid, lines, angles, certain polygons, and theorems involving those.
00:02:30
Speaker
Yeah, and Euclid had like, you know, elements everything is based on a few axioms, very heavily. Right, the postulates. Yeah, it's very heavy postulates as axiom, whatever based system.
Gauss's Mathematical Insights
00:02:44
Speaker
But differential geometry, it doesn't really have axioms per se, right?
00:02:48
Speaker
Well, what we're really thinking of here is where more classical Euclidean geometry is to Euclid. In this case, we're working with differential geometry, which I would say is more of Gauss's realm. And Gauss is an interesting character. Isn't he the one that added up the numbers from 1 to 100? Oh, yeah, that formula.
00:03:11
Speaker
Yeah, he was in second grade and he was annoying his teacher. So she was like, I'd up the numbers from 1 to 100. So he wrote 1 plus 2 plus all the way to 100. And then he wrote it backwards. And they noticed that every entry added up to 101. And he just multiplied that by how many there were and divided by 2 and got the answer in a few minutes.
00:03:32
Speaker
So Gauss did work a lot with surfaces in three-dimensional space, particularly, you know, things like spheres or other curved surfaces, integrals or differentiation among them and theorems relating to such.
00:03:51
Speaker
All right, so what is curvature, basically?
Understanding Curvature with Circles
00:03:55
Speaker
I mean, we could define it in terms of oscillating. They're called oscillating circles. And I think we have a good example for describing kind of what that is.
00:04:08
Speaker
So the concept of describing curvature using oscillating circles can be attributed to Cauchy. And pretty much what that is, is you have, so let's say you have a one-dimensional curve in two-dimensional space. So this could be like, for example, if you're driving on a path, like off-roading or on a road, right? On a flat surface? Or even a 3D surface, right?
00:04:35
Speaker
Yeah, so kind of like a parametric curve is what you're thinking there. Yeah, like you have a point and you have a location is in space over time. So going back to the concept of oscillating circle, so pretty much what we're doing is so at any point on this path, so draw a circle where exactly one point touches that point on the curve.
00:05:01
Speaker
and then just keep expanding the circle until if you expand it any bigger, it would intersect at more than one point. And so that biggest possible circle you can fit into that part of the curve is called the oscillating circle. And the curvature at that point can be described as the reciprocal of that circle's radius.
00:05:29
Speaker
Yeah, so and to clarify too, the circle has to be expanded until it intersects local points, right? Because I was thinking if you have kind of a, imagine if you took a shoelace and you made a kind of a flat oval, but then pulled it back through itself, the oscillating circle would intersect more points on the shoelace than one, right?
00:05:56
Speaker
Right, so we are kind of thinking more on an infinitesimal level. So let's say that we have a short stretch of the plane that is a straight line, then the radius of that osculating circle would technically be infinity because you can just keep growing and growing and it would not ever intersect at that local neighborhood.
00:06:19
Speaker
Yeah, it is a kind of a weird thing that we've been coming back to in breaking math recently. It's describing circles in terms of lines. It's just kind of tangential to that. But so one example that I came up with for describing curvature is talking about like if you're on a car driving on like, let's say a salt flat, right? So if you drive in a circle, you will feel a pull, right? Right. The lateral G's.
00:06:48
Speaker
Yeah, the lateral G is towards the center of the circle. And if you notice that if I'm driving, if I'm driving, let's say like 30 miles an hour, if the circle is really big, like a mile wide, I'll barely feel any acceleration, right? But if it's like a really small circle, like 10 feet wide, I would probably like start to throw up if I'm going 30 miles an hour. Yeah.
00:07:08
Speaker
So you start doing donuts, you're really going to feel it. And that can kind of correspond to curvature is the more lateral G force you feel, the more curvature you would be getting. As you're talking about, like if you're moving along a curve, then you have some amount of acceleration towards the center of, and I guess in this sense, we could call it the oscillating circle. So let's say I'm driving along a weird path in a car, right? So like, let's say I'm tracing out, um, I don't know, a picture of a duck on the salt flat.
00:07:39
Speaker
If I want to know what the curvature is at a certain point on the duck, all I have to do is freeze my hand so that the wheel stays still and the car will go in the oscillating circle. So there's kind of another way to look at it. And then if you measured the acceleration in the car, which can be done by weighing something.
00:07:58
Speaker
All right, so now since we've done curvature, the next part is going to get a little bit into linear algebra. And so before we do that, let's talk about the concepts really quick. All right, Meryl, what's a vector?
00:08:13
Speaker
So there's a number of ways to define a vector. Some people will call it a rank one tensor, or say it's a quantity that has magnitude or direction. But all we really need to know in this case is that it is for our intents and purposes, a collection of real numbers.
00:08:32
Speaker
Yeah, and these numbers through some way, there's many ways to do this, different types of coordinates and whatever. But these numbers describe, in some way, magnitude of direction. All these things are relevant, definitely. Yeah, a vector will have like, let's say we're just in 3D Euclidean space, right? Usually, it'll be the vector x and the vector y and the vector z, right, as the basis.
00:09:00
Speaker
Oh, you're talking about like, you know, orthogonal basics vectors. Okay. Like if we're measuring like space for real, we have to have like, you know, like a point of reference and like, you know, what a unit is and all that. Um, something that I kind of glossed over because I said it was a collection of real numbers. Something that we, I think I need to clarify on is that it is going to be like whatever dimension space is in is how many numbers are in that vector.
00:09:26
Speaker
Oh, yeah. And so, yeah, if you're in four dimensional space, no matter what system you use, you have to define it using four real numbers for linearly independent real numbers.
00:09:37
Speaker
Yeah, that's right. Because if you have two axes pointing in the same direction, then basically like if you're in 3D space, you'll only be able to describe things on either a plane or a line, depending on how linearly dependent the vectors are to each other, right? The basis vectors or the fake basis vectors that don't actually work as basis vectors. Yeah. So it just has to, they have to span the whole space is all that means.
00:10:01
Speaker
Yeah. And I mean, some basis vector type concepts that we use in daily life is like North and South, but also, uh, would you like to describe how Excel, how a velocity, for example, can be described by a vector? So velocity, and this is the high school physics definition is that it's not just speed. Speed is just the magnitude of velocity. Velocity is the speed that something's going at and the direction that it's going.
00:10:27
Speaker
Yeah, so if somebody shoots an arrow, the velocity points along the direction that the arrow was flying, right? Right. And the acceleration can also be described by a vector. And the acceleration is like what you would feel if you were in a carnival ride, like the way that your stomach is being pulled left and right.
00:10:53
Speaker
can be described by acceleration and the direction that you feel the pull is the vector in that situation and the magnitude is how much you feel the pull. So I think that kind of describes vector as well. So next we're going to talk about what matrices are and how they can be used to transform vectors.
00:11:11
Speaker
So let's say to describe what a matrix is, let's say I have a sphere made
Matrices, Vectors, and Eigenvectors
00:11:18
Speaker
out of vectors, right? Like basically vectors pointing in every single direction and they can be described by arrows that are of length one, right?
00:11:26
Speaker
Right. So there's a few things I could do to that sphere. I could stretch it so it turns into an oval, right? I can skew it. I can make it bigger and smaller. Scaling is just stretching equally in all directions, right? So if you want to describe all these things, they're called linear transformations. And they're called linear transformations for one of many reasons. One is that a line multiplied by a matrix will just be another line, right?
00:11:54
Speaker
Right. So for pretty much every normal vector on the sphere that you have, when you're mapping it to something else, there's, you know, a constant scaling associated with it effectively for whichever direction you're pointing in. Yeah. And so you're talking about the direction of the stretching and stuff, right?
00:12:11
Speaker
Right. And, you know, that's why it's called linear is because that scaling is for any one direction. It's constant. It's not, you know, it's not squaring or cubing or, you know, something weird like shiny soil or exponential. Yeah. And also like many linear transformations can be a reversed inverted. So just transformed back. The only times really where they count is when you compress the amount of dimensions they have. Oh, you can't really get that information back. Right.
00:12:40
Speaker
Right. So let's say you have a vector right and you stretch and you stretching the sphere of vectors. If you look at where the original vector was the place where that point is stretched out to the is describes where the vector is when multiplied by that matrix. And then if you the vectors longer than that you just multiply it by how long the vector is divided by one or whatever.
00:13:02
Speaker
because remember the sphere has a radius 1. Exactly. And so also when you stretch this, you'll notice that some vectors might change size, but there's always going to be like, if it's like three-dimensional space, there's usually three. Four-dimensional space is usually four vectors that don't change direction when they're scaled, right?
00:13:22
Speaker
And those are known as eigenvectors. Eigenvectors are a weird concept, but we're going to use them in talking about Gaussian curvature, which is really kind of like a three-dimensional or n-dimensional really description of how things are curved. And eigenvectors and eigenvalues relate to what linear transformations are, correct?
00:13:43
Speaker
That's right, because eigenvectors are the vectors in your domain of your linear transformation that do not in any way get distorted other than scaling by the linear transformation.
00:14:00
Speaker
And what do you mean by domain in this context? I just wanted to clarify the listeners. So let's say that we are just in two dimensional space on the plane, like we have a painting or something and we're applying a linear transformation. So it's going to do some, you know, so it's going to do some stretching and some shearing to it, but there's going to be, um, a particular direction, a particular vector along this painting that doesn't get sheared, just scaled.
00:14:30
Speaker
Yeah. And by the way, sharing, I'm not sure if I described that well earlier. If you want to share a stack of cards, all you have to do is like kind of like put them at an angle, right? Right. So yeah, that's what share. That's what sharing is. I think I called it skewing earlier by accident. But so the eigenvalues is related to the eigenvectors, right?
00:14:51
Speaker
Uh, yeah, because Eigenvalues are just, so for those vectors that I described that only scaling happens along them, uh, Eigenvalues tell you how much scaling happens along those vectors. And what does an Eigenvalue of zero mean? So if you have an Eigenvalue of zero, that means you have a transformation, um, that loses a dimension. So you're compressing. And so you have a linear dependence somewhere probably.
00:15:15
Speaker
Yeah. So let's say we're working in three dimensions, right? And we have two Eigenvalues that are zero. That would mean that we're, we reduce everything to something along the line, correct? Right. And, uh, three Eigenvalues being zero. I'm pretty sure that just means that you're working with a zero matrix, right? Yeah.
00:15:32
Speaker
which is a matrix of all zeros. Oh, and also matrices can be described as like a column, like rows and columns of numbers. So like transformation matrices that transform from like three dimensions to three dimensions would be three wide and three high, right? Exactly.
00:15:51
Speaker
But that's not totally important to what we're talking about today, except all you have to know is that if you want to look at matrix multiplication, you could do it on, Wikipedia has a pretty good article. I mean, essentially it's just you take the rows of the matrix and multiply it by the vector, you multiply it component wise by the vector and add them all up. And then you put all those in a row and that's the product, right?
00:16:19
Speaker
That sounds about right. So let's talk about real quick what the gradient is. The gradient is sort of a way to take the derivative of a vector, right?
00:16:29
Speaker
So the gradient itself is a vector of partial derivatives effectively. And so for any, so let's say that we have a scalar function that takes in a vector, then what we're doing is we're taking the partial derivative with every vector component. And out of that, we're getting a vector
00:16:53
Speaker
of each of those partial derivatives. And that actually tells us. So let's say that we're in two dimensions, just like on a plane. So our scalar function on this plane is going to look like a surface, right? Sort of like a wrinkled up bed sheet, let's say.
00:17:14
Speaker
or like, or you could even say like, uh, so it'd be described the height, right? So you could almost say that you can even describe it as a topological map, right? Or topographic topographic map. Sorry. Yeah. Like one of those mountain maps.
00:17:26
Speaker
Yeah, so what I'm kind of getting at here is that the gradient vector, for one thing, it tells us what at any point the direction of the steepest ascent is, but it also gives us a vector that is tangent to our surface.
00:17:47
Speaker
And so tangent to our surface means that, so if you draw a line in the direction of this vector, it's going to touch locally at exactly one point, and it's going to show us what the slope is at that point. And which direction is the gradient pointing? Because obviously the slope, the gradient could be any, there's a whole circle of vectors that could be tangent to a certain point, right? So which one does the gradient describe?
00:18:17
Speaker
Oh, yeah, that's right. You already said some point of steepest ascent. So yeah, so yeah.
Gradient and Surface Applications
00:18:23
Speaker
So if you have a topological map, the topographic map is probably not the best way to go about it. But if you want to climb a mountain as steep as possible, just take the gradient and then follow it up to the highest point.
00:18:36
Speaker
I bring the gradient up because it's motivation for the tangent vectors on other surfaces, not just these scalar function surfaces, but say like parametric surfaces in three dimensions, three dimensional space. Those will have tangent vectors on them as well. So something that's tangent to like a 2D line, that would be something that points along the line, correct?
00:19:03
Speaker
Uh, yeah. And so we have a one dimensional way of describing this is that a line is tangent to a curve if, so it'll just touch at that one point and it'll share the slope at that point, but not necessarily anywhere else. And so when we're going to surfaces,
00:19:22
Speaker
So we have something similar. We have tangent vectors. So tangent vectors are on a plane that touches a surface. It's tangent to the surface and it can be any vector on that plane. So let's say that we're talking about a human head, right? And you have a book and somebody's rubbing a book on their head. The plane that represents the surface of the book would be tangent to each spot that touches that person's head, correct?
00:19:47
Speaker
Right. So let's say you have a book on someone's head. And so if you were to draw an arrow from the point that the book touches your head just to anywhere on the book, that's a tangent vector. Yeah. And all these tangent vectors together describe something called the tangent space, correct?
00:20:04
Speaker
Right. And so what the tangent space really is, is so for every point on a surface, we have a copy of the plane or n dimensional space that the surface is for that point. So let's say for a sphere, for instance, we just pick any point on it.
00:20:24
Speaker
And so we put a plane tangent to that point. And then so any vector that exists on that plane is part of the tangent space for that point.
00:20:35
Speaker
Yeah, so basically to sum that up, you're saying that every point has a different tangent space. So you could imagine it as like every point on the head has a different orientation of the book. And the points that radiate away on the book from the point that it touches that person's head would represent the tangent space.
00:21:00
Speaker
Yeah, and just a reminder that so the vectors in this tangent space they can have any magnitude and direction they just have to be on that plane. All right, so now let's say somebody uses a pin to pin that book to that person's head. That pin would be normal to the person's head, correct?
00:21:19
Speaker
Right. And so let's say you have any two tangent vectors at a point. So any two vectors that are not the same vector and are not linearly dependent. So let's take a vector that is perpendicular to them that would be pointing outward of the surface. It would be perpendicular to the surface effectively.
00:21:43
Speaker
Yeah, and linearly dependent would mean it's case would mean that they're either pointing in the same direction or the opposite direction. So we just don't want that. Yeah, makes sense. So yeah, that's what the normal of a surface is it just like, who's that? Who's that? Who's that horror movie character with all the nails in his head?
00:22:03
Speaker
Hellraiser apparently. Yeah, so the Hellraiser guy has a bunch of nails tangent, I mean normal to the surface of their skin, right? Exactly. So let's say we want to describe some like some weird surface in three dimensions, right? Like we could do something called parameterization. And parameterization in one dimension could be like a point where it is in time, right?
00:22:26
Speaker
But parametrization in two dimensions is like, let's say that we're doing the surface of somebody's head, right? Again, you can define it in two ways. You could say that the very top of their head to the very bottom of their chin is like an axis.
00:22:44
Speaker
and you could say that the angle that's that like so if they're wearing a fez for example that hat with the tassel the place the tassel would go on their head so the angle that you have to rotate that tassel to to get to the point can be one one parametric variable like u for example
00:23:03
Speaker
or theta or something like that and then v could be or h or whatever could be the height where it is on your head so your eye would be about like you know like five inches down or four inches down from the top of your head and it would be relatively close to zero rotation a little positive or a little negative depending on which eye you were at so that's a way to parameterize the head okay and i just want to interject and say we really seem to be talking a lot about heads today
00:23:30
Speaker
Yeah, I don't know if it's all on your head.
Gaussian Curvature Discussion
00:23:35
Speaker
Okay, so with a lot of big definitions out of the way, now we can finally get to the main course of this episode, and that is Gaussian curvature. And so what that is is, so there's a handful of ways to define it. I will start with the one that says that it's the product of principal curvatures at some point on a surface.
00:24:03
Speaker
So what would be the principle curvature, for example, on a paper towel roll?
00:24:12
Speaker
So a paper towel roll would be a cylinder. So we have, so let's say curving inwards is defined as positive, outwards is negative, just for simplicity sake. So the circular part, so the part that's going around could be considered one direction that we're going in, and that would have a positive curvature.
00:24:34
Speaker
The other that we would have was so along the cylinder going back and forth and so that would be in a straight line and we call that zero curvature.
00:24:44
Speaker
and zero times anything is just zero. And that checks out because if you remember from when we've talked about curvature before, let's say you have a Mylar balloon, you blow it up, but the balloon doesn't stretch. But the curvature of the Mylar balloon doesn't change no matter how it's crumpled up or deflated. In the same way, the curvature of a piece of paper doesn't change if it's
00:25:09
Speaker
if it's crumpled up and or like shaped into whatever. So since you could take a piece of paper and wrap it around a cylinder, it has no curvature.
00:25:21
Speaker
Exactly. So if you have an axis of principal curvature that is zero, that means it's going in a straight line. And that's just what we call zero curvature is a straight line, just like in our original definition of curvature for just a path in a 2D plane.
00:25:42
Speaker
Yeah, and you might be wondering at this stage, like let's say you draw an X on the paper towel roll, it would seem like you have two curvatures multiplying by together with the same sign, right? Which would mean positive curvature. So one way you could find the Gaussian curvature uses what's called a shape operator, right?
00:26:00
Speaker
Yeah, so the shape operator is going to give us our principal curvatures by using eigenvalues and eigenvectors like we talked about. And so what we're taking eigenvalues and eigenvectors of is called the shape operator.
00:26:17
Speaker
Yeah, and the shape operator operates on the normal, the normals right to the surface. And the way that it operates on them is you take the Jacobian. Do you want to describe quickly what the Jacobian is?
00:26:32
Speaker
So the Jacobian is pretty much a vector calculus way of thinking about derivatives is that so if you have a function from three dimensional space to three dimensional space, so you're taking partial derivatives of each component of our function.
00:26:54
Speaker
Yeah, it's like a vector. Remember matrices are like rows of vectors that you each multiply by the vector that you're doing to get the, to get each component right of the new thing. So the shape, so the Jacobian, uh, takes the gradient, um, or the, you know, the method of steepest descent or, uh, basically of each component of the, of the transformation of the function from, uh, that we're describing from like three dimensions to three dimensions or seven to seven or whatever.
00:27:23
Speaker
Yeah, and so what'll happen then is that so each row of a matrix will be one of the gradient vectors for that particular component.
00:27:34
Speaker
Yeah, and what happens if you take the shape operator of the normal at a certain point? So let's say we're doing a cylinder. We actually get a Jacobian that has a row of zero. And that means that the determinant of the Jacobian... Well, let's talk about what that means. It means that...
00:27:58
Speaker
that if you have a sphere, right, that's on the surface, that's like touching the surface, like a bead on the surface of a cylinder, and we multiply it by the Jacobian of the normal at that point, which is the shape operator, right? We're gonna get a flat disk on the surface of the cylinder, and that flat disk has volume zero, which actually corresponds to zero curvature. Basically, your transformations don't change in the up and down direction, or the left and right direction, and then forward to back, you don't actually do anything.
00:28:28
Speaker
Yeah, so let's actually get to our definition of Gaussian curvature in terms of product of principal curvatures. So each principal curvature is an eigenvalue of the shape operator. Yeah, and so in the cylinder you get it that the eigenvalues are 1 and 0 versus if you do a sphere, for example, I mean, in the example that we did, you get the curvature of being 1 over R because it's the determinant of that Jacobian.
00:28:58
Speaker
Right, and so that's the other definition we can use for Gaussian curvature is so we get product of those eigenvalues or principal curvatures, or we get the determinant of the shape operator, they come out to the same thing.
00:29:13
Speaker
And remember the determinant is like the new volume that this like you remember we were talking earlier about transforming the sphere of vectors of the one sphere of vectors. If you if you transform that and you get like a certain different volume that volume divided by the original volume is the determinant. That's how it could be seen is the change in volume.
00:29:34
Speaker
Yeah. So, yeah. So if you have a sphere, for example, so if you have a sphere, the shape operator points into the sphere and then up and down. And the one that points into the sphere is one over R, I believe in magnitude. Right. And because the radius of the sphere is going to be positive, that's going to be positive too.
00:29:55
Speaker
And yeah, so that's how you get the principal curvatures. And yeah, like you said, you multiply them together and you get the Gaussian curvature. How would you apply this to a saddle, for example? So if you just hold it in front of you, then you'll see that going one way, side to side, for instance. So it would be curving up. But from front to back, you'd see that kind of has that instead it curves down. Kind of like a Pringle.
00:30:23
Speaker
Right. So that means, and it doesn't really matter which one you define as negative or positive, because one will be negative, and the other will be positive, and you'll multiply those together and get a surface with negative curvature. Now, why is the curvature of a torus different on the outside and the inside?
00:30:46
Speaker
Okay. So let's say that you pick a point on the outside of a Taurus, a donor chip. Yeah. So it's going to curve in the same direction. So it's going to curve the same way. It's going to curve. We'll call it inward for both. If you go around it horizontally or around it vertically.
00:31:06
Speaker
Yeah, so I mean, like the way you could see that is on the point on the outside, like if you hold your donut, you could hold it so that you're holding it by the side of the donut and your fingers will curve like that. If you rotate your hand, your fingers don't have to curve backwards unnaturally to hold the donut sideways.
00:31:24
Speaker
Right. So let's think about the inside of the donut though. So on your vertical axis again, so it's still curving the same way, but on the horizontal axis, it isn't. It's, I guess, if we call the vertical curving inward, then we would have to call the horizontal curving outward. Yeah. And if you look, if you think about it, if you had a really big donut, you could kind of nestle some, uh, Pringles in there, right? I guess you could, but on the outside, they would stick out all weird.
00:31:52
Speaker
That makes sense, although I'm not sure Pringles and Donuts together are that good a combination. Oh, they're not. Math is full of terrible combinations. And so what can we do with the concept of curvature? What can we apply this to?
00:32:11
Speaker
So one thing that we can do is we can find out what kind of geometry a surface has. We can find out what version of, say, Euclid's parallel postulate that a surface follows. So there's a really elegant version of the parallel postulate that we can use for this.
00:32:30
Speaker
So say we have a line L and a point P that is not on line L, then so what we're used to is Euclidean geometry. So the regular Euclid's parallel postulate that says that we have exactly one line M that P is on that does not intersect with L. So we have one parallel line to L for any point.
00:32:58
Speaker
Yeah. And if I rotate that line at all, right? Like if I changed what line it is, it's going to touch at some point, right? Exactly. Now let's say we're on a sphere. How does this postulate change? So we can think of the, um, so we can think of the lines per se on a sphere as the great circles of the sphere. So all of the circumferences. Yeah. The great circle is the biggest circle that could fit around the sphere, right?
00:33:27
Speaker
Yeah. So that's the thing is that if you have one of these and you have a point not on it, you try to draw the biggest circle around and you're going to intersect at that point. So for spherical or elliptic geometry, there are no parallel lines. This is called the rubber band ball principle. Just kidding. It's not, but it should be. And then finally
Exploring Different Geometries
00:33:48
Speaker
we get honestly, the best part of geometry. If you ask me is hyperbolic geometry.
00:33:57
Speaker
So hyperbolic geometry tells us that if there is a point P, not on line L, then there are infinitely many lines not intersecting L that go through P. Now, there are lines that do intersect L, right?
00:34:15
Speaker
Oh yeah, absolutely. It's just that there's infinitely many that don't. So how does this relate to the curvature of a surface? Well, something that we know about these different geometries is, so how many degrees does a triangle add up to in Euclidean geometry?
00:34:35
Speaker
Um, half of circle. Right. So 180. Yeah. So we can think of what curvature is doing, whether it's negative or positive is giving us a deviation from 180 because a consequence of these parallel postulates of elliptic and hyperbolic geometry is that the angles of a triangle in say elliptic geometry will always add up to more than 180 degrees.
00:35:05
Speaker
So an example of this is if I go to the, if I go to like the equator, like Panama, and I walk or I hovercraft all the way up to the North Pole, and then I hovercraft down to, I don't know, probably somewhere in Africa. Like Equatorial Guinea. Yeah, Equatorial Guinea. It has Equatorial in the name. Yeah. And then back to the place near Nicaragua or whatever we were. I guess that would be Ecuador. Oh, yeah. Again, yeah. Equator name places.
00:35:33
Speaker
Yeah, I forgot about that. But yeah, so you basically turn right three times, which is 270 degrees. Yeah, or near abouts there. And so what's cool about this is positive curvature, positive deviation from 180 degrees.
00:35:52
Speaker
and negative curvature negative deviation. Yeah. So in fact, in some extreme cases of hyperbolic geometry, you can actually get a triangle where all three angles and this is called an asymptotic triangle. All three angles are zero degrees and you have a zero degree triangle. So I mentioned the zero degree triangle, but usually what we get is just a saddle surface. And so that's going to have some negative curvature, not necessarily enough for us to have that
00:36:22
Speaker
asymptotic triangle, but if you try to draw lines, so geodesics along, which are the lines of shortest distance, um, along the surface, then you can use them to get a triangle with angles that add up to less than 180 degrees. So now the, well, the other one was the main course, right? So this is what the dessert.
00:36:49
Speaker
So this is what I was really looking forward to mentioning on this episode is Gauss's Theorem aegregium. And so that's just Latin for remarkable theorem. So let's say that we just take any surface, say, a piece of paper, and so we bend it however we want. We don't stretch it at all. We just bend it. We make sure that all of the metrics on it stay the same effectively.
00:37:17
Speaker
then the curvature on it doesn't change even though it's locally curved right you're saying that the intrinsic curvature doesn't change because like if you wrapped it around something you could say that you know there if you follow a path along the line
00:37:30
Speaker
It would be curved, but the intrinsic curvature, therefore, it's such a remarkable thing, you know, because it could be measured no matter how it's embedded. I mean, you could also think of this as like a deflating balloon or deflating anything like that. Like specifically a balloon that doesn't stretch, like a Mylar balloon or like a balloon for like a, what do you call those balloons that people ride in? Just a hot air balloon.
00:37:56
Speaker
I should know this over an Albuquerque. I lose 10 Albuquerque points, 10 green chilies. But yeah, when this deflates, the curvature does not change just because it's deflated. Because if you think about it, like if I have a globe or an orange peel, I can't really flatten an orange peel onto a table without basically cracking it everywhere, right? Yeah. And so let's think about that for a second. So an orange is a sphere. Yeah. Usually. Just kidding. Yes.
00:38:24
Speaker
Um, spheroid will topologically, it's a spear, but that means, and we've mentioned before, spheres have positive curvature and they're not very stretchy. At least I hope they aren't. What oranges? The, the, the peel on them shouldn't be. Oh yeah. No, you're, you're, yeah, that sounds like some black mirror origins.
00:38:43
Speaker
Yeah, so yeah, if you peel it and you're good enough at peeling an orange to keep it in one contiguous piece, and if you try to flatten it out onto a table, it won't really flatten all the way, like you said, without cracking some. And that's because, so if you could flatten it, it would have zero curvature, right?
00:39:04
Speaker
Yes. But if an orange is spherical, it has positive curvature. So you can't without stretching it, turn it from positive curvature to zero curvature. And in the same way, if you tried to squish a saddle onto the ground, it would be like you would have to kind of like push it together almost like it would bunch up like the surface of a saddle. Right.
00:39:25
Speaker
Yeah. So a saddle isn't going to get flattened on the ground either without, you know, probably without seriously damaging it. All right. So we move on from oranges to pizza.
Gauss's Theorem and Pizza Eating
00:39:35
Speaker
Yeah. So this is the fun thing about, so the theorem of a grudgium can be used as a strategy for eating pizza because don't you just hate it when it's like you're trying to eat pizza, but it's like, you don't want to use both hands necessarily to hold it up. Um, but if you hold it with just one hand, then it might drip down. If let's like, if you're holding the crust, right? Yeah. So let's think about what pizza is. So it's just a disc, a flat disc.
00:40:04
Speaker
So pizza has zero curvature, right? But pizza can bend, right? So let's say we have a slice. So we take a slice, still has zero curvature, and then we lift it up just by the crust. So it's going to droop down, right? Yes.
00:40:24
Speaker
Yeah. So that's gravity doing its work. And so what's happening is it's bending, but not stretching. Yeah. At least isn't stretchy. I would hope it not. If it, I would hope not. Otherwise it's back in the oven. Yeah. But anyway, so what happens then if, so let's say that we hold the pizza such that we're bending the crust. Kind of like a New Yorker.
00:40:48
Speaker
Yeah, so just fold the pizza in half kind of like into a calzone shape. Then, so it'll be curved along that axis. So from the crust to the point of the slice. So what's that going to look like then? That's going to, it's going to just be a straight line, right? Yeah. So the pizza will actually hold up. It won't drip down. Yeah. And the reason why it can't drip down is because if it could, it would be a negative curvature.
00:41:18
Speaker
Right, and it would require some sort of stretching of the pizza, which again, put it back in the oven. And stretching in terms of material science means that you're actively deforming or like you're messing with the bonds between things, right? Just change the axis that the curvature is on and because it's zero, it'll make it easier to eat your pizza. So just eat your pizza like a New Yorker.
Episode Summary on Curvature
00:41:45
Speaker
From Euclid to Gauss, circles to saddles, and pizza to parameterization, we have explored the concept of curvature as it relates to complex surfaces. This concept of curvature was essential for developments in physics, such as general relativity, and is an essential component of a differential geometry arsenal.
00:42:03
Speaker
I'm Sophia. And I'm Meryl. And this has been Breaking Math. Uh, it's been a kind of cool episode. Um, it, one thing that actually kind of directly relates to this episode is the poster we keep plugging, $19 and 65 cents, facebook.com slash breaking math podcast. Click on shop. All right. Till next time. Actually, I don't even know. Bruce Pierre was a complex person, but this is a math podcast and he was terrible.